Osnovy teorii Galua.

~3. TI~ompoehauR qUPicy~e~ U AlUMeKoil14

147

rj~e BefAH41HHb a, b (KoopLHamIM tteHTpa ORpyWHC~oTH) H r (ee paMaHYC),
a TaK}Ke al, bl, r, H3BeCTHbI. BbIqtaTaff OJLHO ypaB~eeiie 143 Aipyroro,
floJlytIHM:
(3.3) (al-a)(2x-a- al)+(bl-b,(2y —&- b)-r2+r1L2= 0.!,
T. e. ypaB~eH~e npqmoil, K0od(~i)HiuHeHTblI KOTOpoIro H3BeCTHbi. Peirnag ero,
COBMeCTHO c KaKHM-HH6yJab 113 ypaBHeHKPI (3.2), MUi n1OJ14HM Jal1 HeHi3BeCTHOfl X (wIMH AAR~ y) KB3uapaHoe yparnefleie. TaKHm o6pa3oM KOOPL1HHaTbi
TO4eK nepece'-eHIMfl 3aJuaHIblX OKpyKHQHCTeA4 onlpeaeaHTC5l, KaK KOPHH KBaJIIaTHbMX yp~aBHeHHtI C H3BeCTHbIMH Koqcp43H[1HeHTamH. $
20. HlYCTb 3aJUaHO KBaLLpaTHoe ypaBHeH-He c H3BeCTHbUMH K03p434HiJmeHTamH:
Ax2 + Bx + C = 0.
Erb KOPHH TaKOBbI:
= -B+B-A
2A
He OCTaHaBRTHBaCb Ha rem~lqC~ nOCTpoeHHH pa1U4H0HanEHb1X Bblpa)KeH11f (Harnptimep JaealeHfe mo}KeT 6binb ocyLIueCTBJaeHo npH nQMOH4Hl Teopembi
o npoIIopLHOHaJ~bHOCTH 0Tpe3KOB mewayU [IapaxlIe~lbHbIMHi npHmbimH), nepeftaem
K noCTpoeHHIO Bb1pa)KeHH~l VB2- 4AC. fHOJUKOjpeHFIoe BbpaweffHHe mower
6bITb npeLC~aBJne~lO KaK npol3BeieHiie P Q I1BYX OTpe3KOB. OTKAaJ~brnasl Ha
npOH3BoJ~bHoil npflmofl PRAJM P H Q, CTP0OH Ha 0Tpe3Ke P + Q. KaK Ha,A~a~e~pe nOJIYOKpY)KHOCTb H BOCCTaBJIIIB 113 TO04K14 pa~zteja 060HX OTpe3KOB
nepneHAHX1KYARiP K HauHIef rlpsimof4, mbI y6eatiMCqH, 'ITO XJUIHHa 9TOFO IeprneH)kHKyA11pa paB~a VP Q.Q
OTmeTHM, 'ITO KBaJUpaTHb1e KOPHH 113 MHHMbIX Bblpa)KeHHfl MOrYT B CHJ1Y
P11a38l III, ~ 3.14 6bI]Tb flpeJLCTaBJIeHbl., KaK BeJJWJHflbl, y KOTOpbIX H4 BetLeCTBeHHbie H MHHmbie qaCTH qBJI5110TC51 KBaJIpaTHbIMH KOPHRiMH 113 BeILxCTBeHHbIX
BeA H'IHH.
2. Hcxoxg 113 Teopembi 74, MLZ BlHJ1HM, I-ITO JUJ1I Toro, 'ITO6bi KOPHH
ypaBHeHH5] (3.1) 6bu~lo B03MO}KHO nohjjyqHTb nOCTpoeHHem nIocpeJCTBOM UH1PKyng1 H JlHHeAKH, HeofxoaiHmo H JLOCTaToMHO, 4TO6bi 9TO ypaBHeH11e nfliBOAHWTOCb K LLeHH KBaJIpaTHbIX ypaBHeHNfl. FIOBTOqgpaHPccyNKILeHH1 Teopembi 68,
MWb! YBHJLHM, WIO Jal1 3Toro rpyH ia IFaJya ypaBHeHHSI (3.1) RiOJrIWia HmeTb,
KOMHO3HLUHOHHblfl PRIA, y KOTOporo pHIL HHJUeKCOB COCTOHT 113 UBoeK. MOKHO
OJUHako JOKa3aTb HeCKOJ~bKo 6onblne, eCJ1H TOJ~bKo npeJUBapHTe.TbHO JUOnOJHHTb CBOH cBeILeHH1I no o6aieg TeopHH rpynii.
3. BO3bmem fpOH3BOJlbH~1O KOHe4HYIO rpyriny ($ H B Hen~ HpOH3BOJlbHb1lA
5AnemeHT S. ByJaem Ha3bIBaTb COBOKYI7H-OCTb BJnemeHTOB, co[np1DKeHHiIX C S,
T. e. nepexo~~tqulx 113 S ripH~ IomoWji4 Hpeo6pa3OBaHHII anemeMHTMH rpynnu~ ($,
K a ac om g.iiemeHra S. TaKHM o6pa3oM Bce sjiemeH b1 rpynriu ($3 MO)KH(Q
pa36HTb Ha KJnaccbl.
npH 9TOM HmeeT MeCTO
Teopema 75. 4HHCTTO KJaCCOB BCAfKofl nOJCTaHOBKH eCTb AeaJHTeIab
rp~yflb, B KOTOPY10 9Ta nOJUCTaHOBKa BXOJUHT.
t 0 K a 3 a T e Ai b C T B 0 CTPOHITCH TaK wKe, KaK JUOI il3aTeJlbCTBO Teopembi if
COBOKyFIHOCTb qJiemeHTOB N rpyrnnbi An, l11 KOTOPEJX HmeeT 7
lo*